REGULARIZATION OF NONLINEAR ILL-POSED
EQUATIONS WITH ACCRETIVE OPERATORS
YA.I.ALBER,C.E.CHIDUME,ANDH.ZEGEYE
Received 11 October 2004
We study the regularization methods for solving equations with arbitrary accretive op-
erators. We establish the strong convergence of these methods and their stability with
respect to perturbations of operators and constraint sets in Banach spaces. Our research
is motivated by the fact that the fixed point problems with nonexpansive mappings are
namely reduced to such equations. Other important examples of applications are evolu-
tion equations and co-variational inequalities in Banach spaces.
1. Introduction
Let Ebe a real normed linear space with dual E∗.Thenormalized duality mapping j:E→
2E∗is defined by
j(x):=x∗∈E∗:x,x∗=x2,
x∗
∗=x, (1.1)
where x,φdenotes the dual product (pairing) between vectors x∈Eand φ∈E∗.It
is well known that if E∗is strictly convex, then jis single valued. We denote the single
valued normalized duality mapping by J.
AmapA:D(A)⊆E→2Eis called accretive if for all x,y∈D(A) there exists J(x−y)∈
j(x−y)suchthat
u−v,J(x−y)≥0, ∀u∈Ax,∀v∈Ay. (1.2)
If Ais single valued, then (1.2)isreplacedby
Ax −Ay,J(x−y)≥0.(1.3)
Ais called uniformly accretive if for all x,y∈D(A) there exist J(x−y)∈j(x−y)anda
strictly increasing function ψ:R+:=[0,∞)→R+,ψ(0) =0suchthat
Ax −Ay,J(x−y)≥ψx−y.(1.4)
Copyright ©2005 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2005:1 (2005) 11–33
DOI: 10.1155/FPTA.2005.11
12 Nonlinear Ill-posed problems with accretive operators
It is called strongly accretive if there exists a constant k>0suchthatin(
1.4)ψ(t)=kt2.If
Eis a Hilbert space, accretive operators are also called monotone.AnaccretiveoperatorA
is said to be hemicontinuous at a point x0∈D(A) if the sequence {A(x0+tnx)}converges
weakly to Ax0for any element xsuch that x0+tnx∈D(A), 0 ≤tn≤t(x0)andtn→0,
n→∞. An accretive operator Ais said to be maximal accretive if it is accretive and the
inclusion G(A)⊆G(B), with Baccretive, where G(A)andG(B) denote graphs of Aand
B, respectively, implies that A=B. It is known (see, e.g., [14]) that an accretive hemicon-
tinuous operator A:E→Ewith a domain D(A)=Eis maximal accretive. In a smooth
Banach space, a maximal accretive operator is strongly-weakly demiclosed on D(A). An
accretive operator Ais said to be m-accretive if R(A+αI)=Efor all α>0, where Iis the
identity operator in E.
Interest in accretive maps stems mainly from their firm connection with fixed point
problems, evolution equations and co-variational inequalites in a Banach space (see, e.g.
[6,7,8,9,10,11,12,26]). Recall that each nonexpansive mapping is a continuous ac-
cretive operator [7,19]. It is known that many physically significant problems can be
modeled by initial-value problems of the form (see, e.g., [10,12,26])
x′(t)+Ax(t)=0, x(0) =x0, (1.5)
where Ais an accretive operator in an appropriate Banach space. Typical examples where
such evolution equations occur can be found in the heat, wave, or Schr¨
odinger equations.
One of the fundamental results in the theory of accretive operators, due to Browder [11],
states that if Ais locally Lipschitzian and accretive, then Ais m-accretive. This result was
subsequently generalized by Martin [23] to the continuous accretive operators. If x(t)in
(1.5)isindependentoft,then(1.5) reduces to the equation
Au =0, (1.6)
whose solutions correspond to the equilibrium points of the system (1.5). Consequently,
considerable research efforts have been devoted, especially within the past 20 years or so,
to iterative methods for approximating these equilibrium points.
The two well-known iterative schemes for successive approximation of a solution of
the equation Ax =f,whereAis either uniformly accretive or strongly accretive, are the
Ishikawa iteration process (see, e.g., [20]) and the Mann iteration process (see, e.g., [22]).
These iteration processes have been studied extensively by various authors and have been
successfully employed to approximate solutions of several nonlinear operator equations
in Banach spaces (see, e.g., [13,15,17]). But all efforts to use the Mann and the Ishikawa
schemes to approximate the solution of the equation Ax =f,whereAis an accretive-type
mapping (not necessarily uniformly or strongly accretive), have not provided satisfactory
results. The major obstacle is that for this class of operators the solution is not, in general,
unique.
Our purpose in this paper is to construct approximations generated by regularization
algorithms, which converge strongly to solutions of the equations Ax =fwith accretive
maps Adefined on subsets of Banach spaces. Our theorems are applicable to much larger
classes of operator equations in uniformly smooth Banach spaces than previous results
Ya . I . A l b e r e t a l . 1 3
(see, e.g., [4]). Furthermore, the stability of our methods with respect to perturbation of
the operators and constraint sets is also studied.
2. Preliminaries
Let Ebe a real normed linear space of dimension greater than or equal to 2, and x,y∈E.
The modulus of smoothness of Eis defined by
ρE(τ):=supx+y+x−y
2−1:x=1, y=τ.(2.1)
ABanachspaceEis called uniformly smooth if
lim
τ→0hE(τ):=lim
τ→0
ρE(τ)
τ=0.(2.2)
Examples of uniformly smooth spaces are the Lebesgue Lp, the sequence ℓp, and the
Sobolev Wm
pspaces for 1 <p<∞and m≥1 (see, e.g., [2]).
If Eis a real uniformly smooth Banach space, then the inequality
x2≤y2+2x−y,Jx
≤y2+2x−y,Jy+2x−y,Jx−Jy(2.3)
holds for every x,y∈E. A further estimation of x2needs one of the following two
lemmas.
Lemma 2.1 [5]. Let Ebe a uniformly smooth Banach space. Then for x,y∈E,
x−y,Jx−Jy≤8x−y2+Cx,yρEx−y, (2.4)
where
Cx,y≤4max2L,x+y(2.5)
and Lis the Figiel constant, 1<L<1.7[
18,24].
Lemma 2.2 [2]. In a uniformly smooth Banach space E,forx,y∈E,
x−y,Jx−Jy≤R2x,yρE4x−y
Rx,y, (2.6)
where
Rx,y=2−1x2+y2.(2.7)
If x≤Rand y≤R, then
x−y,Jx−Jy≤2LR2ρE4x−y
R, (2.8)
where Lis the same as in Lemma 2.1.
14 Nonlinear Ill-posed problems with accretive operators
We will need the following lemma on the recursive numerical inequalities.
Lemma 2.3 [1]. Let {λk}and {γk}be sequences of nonnegative numbers and let {αk}be a
sequence of positive numbers satisfying the conditions
∞
1
αn=∞,γn
αn−→ 0as n−→ ∞ .(2.9)
Let the recursive inequality
λn+1 ≤λn−αnφλn+γn,n=1,2, ..., (2.10)
be given where φ(λ)is a continuous and nondecreasing function from R+to R+such that it
is positive on R+\{0},φ(0) =0,limt→∞ φ(t)≥c>0. Then λn→0as n→∞.
We will also use the concept of a sunny nonexpansive retraction [19].
Definition 2.4. Let Gbe a nonempty closed convex subset of E.AmappingQG:E→Gis
said to be
(i) a retraction onto Gif Q2
G=QG;
(ii) a nonexpansive retraction if it also satisfies the inequality
QGx−QGy
≤x−y,∀x,y∈E; (2.11)
(iii) a sunny retraction if for all x∈Eand for all 0 ≤t<∞,
QGQGx+tx−QGx=QGx. (2.12)
Definition 2.5. If QGsatisfies (i)–(iii) of Definition 2.4, then the element x=QGxis said
to be a sunny nonexpansive retractor of x∈Eonto G.
Proposition 2.6. Let Ebe a uniformly smooth Banach space, and let Gbeanonempty
closed convex subset of E. A mapping QG:E→Gis a sunny nonexpansive retraction if and
only if for all x∈Eand for all ξ∈G,
x−QGx,JQGx−ξ≥0.(2.13)
Denote by ᏴE(G1,G2) the Hausdorffdistance between sets G1and G2in the space E,
that is,
ᏴEG1,G2=maxsup
z1∈G1
inf
z2∈G2
z1−z2
,sup
z1∈G2
inf
z2∈G1
z1−z2
.(2.14)
Lemma 2.7 [7]. Let Ebe a uniformly smooth Banach space, and let Ω1and Ω2be closed
convex subsets of Esuch that the Hausdorffdistance ᏴE(Ω1,Ω2)≤σ.IfQΩ1and QΩ2are
the sunny nonexpansive retractions onto the subsets Ω1and Ω2,respectively,then
QΩ1x−QΩ2x
2≤16R(2r+q)hE16LR−1σ, (2.15)
where hE(τ)=τ−1ρE(τ),Lis the Figiel constant, r=x,q=max{q1,q2},andR=2(2r+
q)+σ.Hereqi=dist(θ,Ωi),i=1,2,andθis the origin of the space E.
Ya . I . A l b e r e t a l . 1 5
3. Operator regularization method
We will deal with accretive operators A:E→Eand operator equation
Ax =f(3.1)
given on a closed convex subset G⊂D(A)⊆E,whereD(A)isadomainofA.
In the sequel, we understand a solution of (3.1) in the sense of a solution of the co-
variational inequality (see, e.g., [9])
Ax −f,J(y−x)≥0, ∀y∈G,x∈G. (3.2)
The following statement is a motivation of this approach [25].
Theorem 3.1. Suppose that Eis a reflexive Banach space with strictly convex dual space E∗.
Let A:E→Ebeahemicontinuousoperator.Ifforfixedx∗∈Eand f∈Ethe co-variational
inequality
Ax −f,Jx−x∗≥0, ∀x∈E, (3.3)
holds, then Ax∗=f.
In fact, the following more general theorem was proved in [8].
Theorem 3.2. Let Ebe a smooth Banach space and let A:E→2Ebe an accretive operator.
Then the following statements are equivalent:
(i) x∗satisfies the covariational inequality
z−f,Jx−x∗≥0, ∀z∈Ax,∀x∈E; (3.4)
(ii) 0 ∈R(Ax∗−f).
We present the following two definitions of a solution of the operator equation (3.1)
on G.
Definition 3.3. An element x∗∈Gis said to be a generalized solution of the operator
equation (3.1)onGif there exists z∈Ax∗such that
z−f,Jy−x∗≥0, ∀y∈G. (3.5)
Definition 3.4. An element x∗∈Gis said to be a total solution of the operator equation
(3.1)onGif
z−f,Jy−x∗≥0, ∀y∈G,∀z∈Ay. (3.6)
Lemma 3.5 [6]. Suppose that Eis a reflexive Banach space with strictly convex dual space
E∗.LetAbe an accretive operator. If an element x∗∈Gis the generalized solution of (3.1)
on Gcharacterized by the inequality (3.5), then it satisfies also the inequality (3.6), that is,
it is a total solution of (3.1).